We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here, we consider polynomials of the form $P\left(x\right)=\sum _{k=0}^d{x}^k{a}_k$, with Clifford coefficients $a_k\in {\mathbb{R}}_n$, and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy–Riemann) operator and extend the results to slice-regular functions.

中文翻译：

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Clifford 代数上切片正则函数的 Almansi 型定理

我们提出了具有 Clifford 系数的多项式的 Almansi 型分解，更一般地用于 Clifford 代数上的切片正则函数。埃米利奥·阿尔曼西 (Emilio Almansi) 于 1899 年发表的经典结果处理了多谐函数，即迭代拉普拉斯算子的核元素。在这里，我们考虑以下形式的多项式$磷\left(X\right)=\sum _{克=0}^d{X}^{克}{\mathrm{一种}}_{克}$, 与 Clifford 系数 ${\mathrm{一种}}_{克}\in {\mathbb{电阻}}_n$，并得到与区域多谐函数相关的类似分解。我们展示了这种分解与狄拉克（或柯西-黎曼）算子之间的关系，并将结果扩展到切片正则函数。